Determination of Local Stresses and Strains within the Notch ...

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applied sciences Article Determination of Local Stresses and Strains within the Notch Strain Approach: Efficient Implementation of Notch Root Approximations Ralf Burghardt * , Lukas Masendorf, Michael Wächter and Alfons Esderts Citation: Burghardt, R.; Masendorf, L.; Wächter, M.; Esderts, A. Determination of Local Stresses and Strains within the Notch Strain Approach: Efficient Implementation of Notch Root Approximations. Appl. Sci. 2021, 11, 10339. https://doi.org/ 10.3390/app112110339 Academic Editor: Filippo Berto Received: 6 October 2021 Accepted: 2 November 2021 Published: 3 November 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). Institute for Plant Engineering and Fatigue Analysis, Clausthal University of Technology, Leibnizstraße 32, 38678 Clausthal-Zellerfeld, Germany; [email protected] (L.M.); [email protected] (M.W.); [email protected] (A.E.) * Correspondence: [email protected] Abstract: An estimation of the elastic-plastic stress state using elasticity-theoretical input data is an essential part of the service life estimation with the local strain approach in general and a German guideline based on it, in particular. This guideline uses two different notch root approximations (an extended version of Neuber’s rule and an approach according to Seeger and Beste) for this estimation. Both require the implementation of Newton’s method to be iteratively solved. However, many options are left open to the user concerning implementation in program code. This paper discusses ways in which notch root approximation methods can be implemented efficiently for use in software systems and elaborates an application recommendation. The following aspects and their influence on the computational accuracy and performance of Newton’s method are considered in detail: influence of the formulation of the root finding problem, determination of the derivative required for Newton’s method and influence of the termination criterion. The investigation shows that the advice given in the abovementioned guideline indeed leads to a conservative implementation. By carefully considering the investigated aspects, however, the computational performance can be increased by approximately a factor of 2–3 without influencing the accuracy of the service life estimation. Keywords: Neuber’s rule; fatigue life calculation; notch strains; notch stresses; fatigue of materials; local strain approach; notch strain concept 1. Introduction The assessment of fatigue strength is one of the most important aspects in the design of safety-relevant components. As an alternative or a supplement to an experimental strength assessment, an analytical assessment can be performed. Various calculation approaches exist for this purpose, which can be distinguished into stress- and strain-based concepts [1,2]. The latter, which are also called local strain approaches, assume elastic- plastic material behavior in the component and use the calculated local stresses and strains as basic load quantities. The notch strain concept, also known as the local strain approach, can be found in many variants throughout the literature [15]. These variants differ primarily in the procedures used to determine the local stresses and strains, e.g., the notch root approx- imations [616], and in the load parameters used to evaluate the damage of individual stress-strain hystereses, [1724]. Both can have a significant influence on the calculation result of the component lifetime. In addition, the calculation procedures of the notch strain approach—regardless of how it is expressed—can only be applied with the help of computational algorithms, since, for example, numerical solution procedures need to be used. Therefore, the calculation result also depends to a certain extent on the user-specific implementation of these calculation algorithms. Appl. Sci. 2021, 11, 10339. https://doi.org/10.3390/app112110339 https://www.mdpi.com/journal/applsci

Transcript of Determination of Local Stresses and Strains within the Notch ...

applied sciences

Article

Determination of Local Stresses and Strains within the NotchStrain Approach: Efficient Implementation of NotchRoot Approximations

Ralf Burghardt * , Lukas Masendorf, Michael Wächter and Alfons Esderts

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Citation: Burghardt, R.; Masendorf,

L.; Wächter, M.; Esderts, A.

Determination of Local Stresses and

Strains within the Notch Strain

Approach: Efficient Implementation

of Notch Root Approximations. Appl.

Sci. 2021, 11, 10339. https://doi.org/

10.3390/app112110339

Academic Editor: Filippo Berto

Received: 6 October 2021

Accepted: 2 November 2021

Published: 3 November 2021

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Copyright: © 2021 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Institute for Plant Engineering and Fatigue Analysis, Clausthal University of Technology, Leibnizstraße 32,38678 Clausthal-Zellerfeld, Germany; [email protected] (L.M.);[email protected] (M.W.); [email protected] (A.E.)* Correspondence: [email protected]

Abstract: An estimation of the elastic-plastic stress state using elasticity-theoretical input data is anessential part of the service life estimation with the local strain approach in general and a Germanguideline based on it, in particular. This guideline uses two different notch root approximations (anextended version of Neuber’s rule and an approach according to Seeger and Beste) for this estimation.Both require the implementation of Newton’s method to be iteratively solved. However, manyoptions are left open to the user concerning implementation in program code. This paper discussesways in which notch root approximation methods can be implemented efficiently for use in softwaresystems and elaborates an application recommendation. The following aspects and their influenceon the computational accuracy and performance of Newton’s method are considered in detail:influence of the formulation of the root finding problem, determination of the derivative required forNewton’s method and influence of the termination criterion. The investigation shows that the advicegiven in the abovementioned guideline indeed leads to a conservative implementation. By carefullyconsidering the investigated aspects, however, the computational performance can be increased byapproximately a factor of 2–3 without influencing the accuracy of the service life estimation.

Keywords: Neuber’s rule; fatigue life calculation; notch strains; notch stresses; fatigue of materials;local strain approach; notch strain concept

1. Introduction

The assessment of fatigue strength is one of the most important aspects in the designof safety-relevant components. As an alternative or a supplement to an experimentalstrength assessment, an analytical assessment can be performed. Various calculationapproaches exist for this purpose, which can be distinguished into stress- and strain-basedconcepts [1,2]. The latter, which are also called local strain approaches, assume elastic-plastic material behavior in the component and use the calculated local stresses and strainsas basic load quantities.

The notch strain concept, also known as the local strain approach, can be foundin many variants throughout the literature [1–5]. These variants differ primarily in theprocedures used to determine the local stresses and strains, e.g., the notch root approx-imations [6–16], and in the load parameters used to evaluate the damage of individualstress-strain hystereses, [17–24]. Both can have a significant influence on the calculationresult of the component lifetime. In addition, the calculation procedures of the notchstrain approach—regardless of how it is expressed—can only be applied with the help ofcomputational algorithms, since, for example, numerical solution procedures need to beused. Therefore, the calculation result also depends to a certain extent on the user-specificimplementation of these calculation algorithms.

Appl. Sci. 2021, 11, 10339. https://doi.org/10.3390/app112110339 https://www.mdpi.com/journal/applsci

Appl. Sci. 2021, 11, 10339 2 of 23

To be able to use local strain approaches for a reliable analytical component assess-ment, the abovementioned diversity of variants and the dependency of the calculationresult on the implementation tend to be disadvantages. Calculation results are not com-parable. To overcome this weakness, the guideline “Rechnerischer Festigkeitsnachweisfür Maschinenbauteile unter expliziter Erfassung nichtlinearen Werkstoffverformungsver-haltens” (Analytical strength assessment for components under explicit consideration ofnonlinear material behaviour, [25]) has been developed in Germany. In this guideline, theabovementioned diversity of variants in the calculation algorithms has been reduced totwo variants of different complexity. Both variants are also described in such detail thatthe deviations in the calculation results by different users are reduced to a minimum. Inaddition, the calculation algorithm is provided with a safety factor concept that allowsthe assessment of component fatigue lives for low probabilities of failure. The describedguideline was developed by the German research association “ForschungskuratoriumMaschinenbau” (FKM) and will be referred to as the FKM Guideline nonlinear in thefollowing. As this guideline is only available in the German language so far, the calculationmodel for the fatigue strength assessment will be briefly explained in the following. Moredetailed information can be found in [25,26].

The FKM Guideline nonlinear allows two different calculation procedures for fatiguestrength assessment, which differ in the estimation of the local stresses and strains and theload parameter used.

• On the one hand, the local elastic-plastic stresses and strains can be estimated usingthe notch root approximation according to Neuber [27] with the extension according toSeeger and Heuler [28]. For each closed hysteresis detected from the local stress-strainpaths estimated in this way, a load parameter PRAM, Equation (1), is calculated.

PRAM =√(σa+k · σm) · εa · E if (σa+k · σm) ≥ 0

PRAM = 0 if (σa+k · σm) < 0(1)

Here, σa is the stress amplitude, σm is the mean stress and εa is the strain amplitudeof the detected hysteresis. E is Young’s modulus, and k is a correction factor that takes intoaccount the mean stress sensitivity of the material.

The load parameter PRAM is a modification of the widely used approach accordingto Smith, Watson and Topper [17]. This modification extends the Smith, Watson, Topperapproach by a material-dependent mean stress sensitivity, as originally suggested byBergmann [19,29].

With the help of the PRAM values for the individual hystereses and a correspondingPRAM Wöhler curve, damage accumulation can be carried out, and the component servicelife can be calculated.

• On the other hand, the local stresses and strains can be estimated using the notch rootapproximation according to Seeger and Beste [30–32]. The evaluation of the damageof the individual hystereses is carried out with the PRAJ load parameter, which is arefined version of the PJ parameter according to Vormwald [21,24].

PRAJ= 1.24 · ∆σeff2

E+

1.02√n′· ∆σeff · ∆εeff −

∆σeffE

(2)

This is motivated by the fracture mechanics of short cracks and is thus able to takeinto account the sequence effects of different load sequences through crack opening andclosing effects.

The calculation procedure with PRAM is less complex in terms of the calculationalgorithm and therefore is easier to implement. However, the variant using PRAJ has theadvantage that load sequence effects can be taken into account, which promises a betteraccuracy, especially for load sequences with changing statistical characteristics.

Both described calculation procedures have in common that a notch root approxima-tion is used to determine the local stresses and strains (extended notch root approximation

Appl. Sci. 2021, 11, 10339 3 of 23

according to Neuber or notch root approximation according to Seeger and Beste). Withthese, the local elastic-plastic stresses are estimated based on stresses that were determinedusing linear-elastic material behavior and, for example, finite element (FE) analysis. Theprocedure for a notch root approximation is shown in a generalized and simplified form inFigure 1. The determination of the local linear-elastic stress is marked as point 1.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 24

Both described calculation procedures have in common that a notch root approximation is used to determine the local stresses and strains (extended notch root approximation according to Neuber or notch root approximation according to Seeger and Beste). With these, the local elastic-plastic stresses are estimated based on stresses that were determined using linear-elastic material behavior and, for example, finite element (FE) analysis. The procedure for a notch root approximation is shown in a generalized and simplified form in Figure 1. The determination of the local linear-elastic stress is marked as point 1.

Figure 1. Estimation of local elastic-plastic stresses and strains from linear-elastic stresses using a notch root approximation (schematically).

The notch root approximation determines how linear-elastic and elastic-plastic stresses and strains are related. Thus, starting from point 1, the intersection with the elastic-plastic material law can be determined with the help of the notch approximation function (point 2). The combination of elastic-plastic stress and strain determined in this way represents the estimate for the local stress-strain state.

The specific implementation for the two notch root approximations in question, which rely on numerical solution methods, is not completely covered within the FKM Guideline nonlinear [25].

Therefore, in this paper, the focus is on the application of notch approximation methods. On the one hand, the correctness of the implementation and, on the other hand, the efficiency with respect to the computation time of different implementation variants shall be demonstrated.

This paper is structured as follows: Section 2 describes the notch root approximations used in the FKM Guideline nonlinear. In Section 3, issues regarding the implementation of notch root approximations are listed and explained. These issues result in several possible implementations of the notch root approximations in program code. Therefore, in Section 4, some of these variants are implemented, evaluated, and discussed with regard to the expected performance and accuracy. For this purpose, a database of artificial geometries in combination with different configurations for the cyclic material properties is used. Finally, Section 5 summarizes the results with recommendations for application.

2. Notch Root Approximations in the FKM Guideline Nonlinear Two different notch root approximations are available in the calculation algorithm of

the FKM Guideline nonlinear: • Notch root approximation according to Neuber [27] in the variant according to [28]. • Notch root approximation according to Seeger and Beste [30–32].

Both notch root approximations are used in the FKM Guideline nonlinear in such a way that they establish the relationship between a linear-elastically determined stress and the local elastic-plastic stresses and strains. The elastic-plastic stress state is thus

(εel,σel)

Hook

Strain ε

Stre

ssσ (εel,pl,σel,pl)

Elastic-plastic material behaviour

1

❌2

Notch rootapproximation❌

Figure 1. Estimation of local elastic-plastic stresses and strains from linear-elastic stresses using anotch root approximation (schematically).

The notch root approximation determines how linear-elastic and elastic-plastic stressesand strains are related. Thus, starting from point 1, the intersection with the elastic-plasticmaterial law can be determined with the help of the notch approximation function (point 2).The combination of elastic-plastic stress and strain determined in this way represents theestimate for the local stress-strain state.

The specific implementation for the two notch root approximations in question, whichrely on numerical solution methods, is not completely covered within the FKM Guidelinenonlinear [25].

Therefore, in this paper, the focus is on the application of notch approximation meth-ods. On the one hand, the correctness of the implementation and, on the other hand, theefficiency with respect to the computation time of different implementation variants shallbe demonstrated.

This paper is structured as follows: Section 2 describes the notch root approximationsused in the FKM Guideline nonlinear. In Section 3, issues regarding the implementation ofnotch root approximations are listed and explained. These issues result in several possibleimplementations of the notch root approximations in program code. Therefore, in Section 4,some of these variants are implemented, evaluated, and discussed with regard to theexpected performance and accuracy. For this purpose, a database of artificial geometries incombination with different configurations for the cyclic material properties is used. Finally,Section 5 summarizes the results with recommendations for application.

2. Notch Root Approximations in the FKM Guideline Nonlinear

Two different notch root approximations are available in the calculation algorithm ofthe FKM Guideline nonlinear:

• Notch root approximation according to Neuber [27] in the variant according to [28].• Notch root approximation according to Seeger and Beste [30–32].

Both notch root approximations are used in the FKM Guideline nonlinear in such away that they establish the relationship between a linear-elastically determined stress andthe local elastic-plastic stresses and strains. The elastic-plastic stress state is thus estimated.This relationship between the linear-elastic stress and elastic-plastic stress is called theload-notch-strain curve of the component. This is used as a template with discrete valuesfor the respective stresses in the Rainflow HCM algorithm [33] to simulate the path of thelocal stresses and strains depending on the path of the linear-elastic stress (load sequence)

Appl. Sci. 2021, 11, 10339 4 of 23

and to identify closed hysteresis loops. The HCM algorithm takes into account the effectsof Masing’s behavior during load reversal [34] and material memory [1,33].

The FKM Guideline nonlinear uses the approach according to Ramberg and Osgoodto describe the elastic-plastic material behavior [35].

εel,pl =σel,pl

E+

(σel,pl

K′

) 1n′

(3)

Here, E is Young’s modulus, K’ is the cyclic hardening coefficient, and n’ is the cyclichardening exponent.

The determination of these parameters can be carried out either experimentally or byusing estimation methods. For the evaluation of strain-controlled tests, the FKM Guidelinenonlinear suggests a procedure based on [36] and [37]. In addition, a method developedby Wächter et al. [38,39] for the estimation of the cyclic properties using tensile strength isprovided for use. This enables an analytical fatigue assessment to be carried out withoutthe need for expensive experiments.

Using the cyclic parameters K’ and n’, it is possible to calculate a cyclic yield strengthR’p0,2 as in Equation (4). R´p0,2 will later be of importance to estimate a maximum load.

R′p0.2= 0.002n′ ·K′ (4)

For the material behavior on a hysteresis branch, the doubled curve applies withregard to stress and strain because of Masing’s behavior, illustrated in Figure 2:

∆εel,pl =∆σel,pl

E+2·

(∆σel,pl

2 ·K′) 1

n′(5)

Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 24

estimated. This relationship between the linear-elastic stress and elastic-plastic stress is called the load-notch-strain curve of the component. This is used as a template with discrete values for the respective stresses in the Rainflow HCM algorithm [33] to simulate the path of the local stresses and strains depending on the path of the linear-elastic stress (load sequence) and to identify closed hysteresis loops. The HCM algorithm takes into account the effects of Masing’s behavior during load reversal [34] and material memory [1,33].

The FKM Guideline nonlinear uses the approach according to Ramberg and Osgood to describe the elastic-plastic material behavior [35].

εel,pl = σel,pl

E +σel,pl

K’

1n’

(3)

Here, E is Young’s modulus, K’ is the cyclic hardening coefficient, and n’ is the cyclic hardening exponent.

The determination of these parameters can be carried out either experimentally or by using estimation methods. For the evaluation of strain-controlled tests, the FKM Guideline nonlinear suggests a procedure based on [36] and [37]. In addition, a method developed by Wächter et al. [38,39] for the estimation of the cyclic properties using tensile strength is provided for use. This enables an analytical fatigue assessment to be carried out without the need for expensive experiments.

Using the cyclic parameters K’ and n’, it is possible to calculate a cyclic yield strength R’p0,2 as in Equation (4). R p0,2 will later be of importance to estimate a maximum load.

Rp0.2’ = 0.002n’⋅K’ (4)

For the material behavior on a hysteresis branch, the doubled curve applies with regard to stress and strain because of Masing’s behavior, illustrated in Figure 2:

Δεel,pl=Δσel,pl

E +2·Δσel,pl

2⋅K’

1n’

(5)

Figure 2. Schematic illustration of Masing’s behavior (a) and the shape of the stress-strain curve due to Masing’s behavior (b).

Due to this different behavior at the initial load and on load reversal, two load-notch-strain curves have to be determined. This can be performed before applying the HCM algorithm. In the FKM Guideline nonlinear, a total of 100 values with equally spaced intervals between the respective elasticity-theoretical values are specified for the discretization of the load-notch-strain curve for the initial load. According to Equation (5), the number of values for the hysteresis branch must be twice as large and amount to 200. The procedure is shown schematically for only eight values in Figure 3.

Strain ε

Stress σ

Strain Δε, εa

Stress Δσ, σa

x 2

Δσ

Δε

Δσ

Δε

initial load curve

hysteresis branch

(a) (b)

Figure 2. Schematic illustration of Masing’s behavior (a) and the shape of the stress-strain curve dueto Masing’s behavior (b).

Due to this different behavior at the initial load and on load reversal, two load-notch-strain curves have to be determined. This can be performed before applying theHCM algorithm. In the FKM Guideline nonlinear, a total of 100 values with equallyspaced intervals between the respective elasticity-theoretical values are specified for thediscretization of the load-notch-strain curve for the initial load. According to Equation (5),the number of values for the hysteresis branch must be twice as large and amount to 200.The procedure is shown schematically for only eight values in Figure 3.

Appl. Sci. 2021, 11, 10339 5 of 23Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 24

Figure 3. Creation of a discretized load-notch-strain-curve via notch root approximations [40].

The load-notch-strain curves of the component for the initial load and for the hysteresis branch are related. It is possible to derive the initial load-notch-strain curve from the load-notch-strain curve of the hysteresis branch. By halving the values of elastic stress Δσel, elastic-plastic stress Δσel,pl , and elastic-plastic strain Δεel,pl range of every second of the abovementioned interval of the load-notch-strain-curve of the hysteresis branch, the initial load-notch-strain-curve is obtained. This is used in Sections 4 and 5 to limit the investigation to the determination of the load-notch-strain-curve for the hysteresis branch.

For the application of both notch root approximations available in the FKM Guideline nonlinear, the knowledge of the limit load factor Kp is necessary. The limit load factor describes the ratio of the external load Le, which leads to yielding at the local point under consideration, and the external load Lp, which leads to global component collapse in the case of an elastic-ideal plastic material law, Equation (6). The limit load factor can either be approximately calculated by analytical relations for simple geometries [28,31,41] or, in the case of complex geometries, by FE analyses using elastic-ideal plastic material behavior. More details on the determination of Kp can be found in [28]. The limit load factor is assumed to be given from this point on.

Kp=Lp

Le (6)

At first glance, it may seem questionable to perform a notch root approximation if results from an elastic-plastic FE analysis are necessary for its application. However, it should be pointed out that the FE calculation is only needed to determine the load Lp leading to plastic collapse. For this purpose, a much coarser FE mesh can be used compared to what would be necessary for the calculation of precise stress and strain values. In addition, Kp is calculated with elastic-ideal-plastic material behavior. Both lead to significantly less calculation effort than when calculating the load-notch-strain curve using FE analysis with the Ramberg-Osgood material behavior.

2.1. Extended Version of Neuber’s Rule The original version of Neuber’s rule [27] postulates, assuming elastic net-section

behavior, the equality of the products of stress and strain for linear-elastic and elastic-plastic material behavior, Equation (7):

σel,pl ⋅ εel,pl = σel ⋅ εel (7)

As shown by Seeger et al. [28,30–32] but also noted by Ellyin and Kujawski [42], this equation must be modified to account for high loads. To take into account the increasing net-section plasticity, Seeger and Heuler proposed a modification of the Neuber equation from Equation (7):

Δσ e

lΔσ e

l,pl

ΔεelΔεel,pl

notch rootapproximations

template

Δε

Δσ

8

8

7

7

6

65

54

4332

1 21

linear elasticcurve

Elastic-plastic curve

discreteranges

00

Figure 3. Creation of a discretized load-notch-strain-curve via notch root approximations [40].

The load-notch-strain curves of the component for the initial load and for the hysteresisbranch are related. It is possible to derive the initial load-notch-strain curve from theload-notch-strain curve of the hysteresis branch. By halving the values of elastic stress∆σel, elastic-plastic stress ∆σel,pl, and elastic-plastic strain ∆εel,pl range of every secondof the abovementioned interval of the load-notch-strain-curve of the hysteresis branch,the initial load-notch-strain-curve is obtained. This is used in Sections 4 and 5 to limit theinvestigation to the determination of the load-notch-strain-curve for the hysteresis branch.

For the application of both notch root approximations available in the FKM Guidelinenonlinear, the knowledge of the limit load factor Kp is necessary. The limit load factordescribes the ratio of the external load Le, which leads to yielding at the local point underconsideration, and the external load Lp, which leads to global component collapse in thecase of an elastic-ideal plastic material law, Equation (6). The limit load factor can either beapproximately calculated by analytical relations for simple geometries [28,31,41] or, in thecase of complex geometries, by FE analyses using elastic-ideal plastic material behavior.More details on the determination of Kp can be found in [28]. The limit load factor isassumed to be given from this point on.

Kp =Lp

Le(6)

At first glance, it may seem questionable to perform a notch root approximationif results from an elastic-plastic FE analysis are necessary for its application. However,it should be pointed out that the FE calculation is only needed to determine the loadLp leading to plastic collapse. For this purpose, a much coarser FE mesh can be usedcompared to what would be necessary for the calculation of precise stress and strain values.In addition, Kp is calculated with elastic-ideal-plastic material behavior. Both lead tosignificantly less calculation effort than when calculating the load-notch-strain curve usingFE analysis with the Ramberg-Osgood material behavior.

2.1. Extended Version of Neuber’s Rule

The original version of Neuber’s rule [27] postulates, assuming elastic net-sectionbehavior, the equality of the products of stress and strain for linear-elastic and elastic-plasticmaterial behavior, Equation (7):

σel,pl · εel,pl= σel · εel (7)

As shown by Seeger et al. [28,30–32] but also noted by Ellyin and Kujawski [42], thisequation must be modified to account for high loads. To take into account the increasingnet-section plasticity, Seeger and Heuler proposed a modification of the Neuber equationfrom Equation (7):

σel,pl · εel,pl= σel · ε∗el ·Kp (8)

Appl. Sci. 2021, 11, 10339 6 of 23

The use of ε∗el · Kp modifies the linear-elastic material behavior so that on the onehand, the real material behavior and on the other hand, the increasing net-section plasticityare taken into account [28,40]. The net section plasticity is modelled by ε∗el using fictitiousnominal stresses, obtained by means of the linear-elastic local stress σel and the limit loadfactor Kp, and inserting these into the material law. In this case the Ramberg and OsgoodEquation (3) approach is used.

ε∗el =σel/ Kp

E+

(σel/ Kp

K′

)1/n′

(9)

As mentioned above the load-notch-strain curve for initial load can be derived fromthe load-notch-strain curve of the hysteresis branch. Therefore, in the following only theprocedure and the root finding equations for the hysteresis branch are explained. Thedetailed derivation, which also includes the formulas for the initial load curve, is shownin Appendix A. Equation (8) applied to the hysteresis branch can be transformed, asdescribed above, into the root finding problem specified in the FKM Guideline nonlinearEquation (10).

f(∆σ el,pl) = 0 =∆σel,pl

E+2 ·

(∆σel,pl

2 ·K′)1/n′

−(

∆σel∆σel,pl

·Kp ·(

∆σel/Kp

E+2 ·

(∆σel/Kp

2 ·K′)1/n′

))(10)

Equation (10) has to be solved for the 200 discrete required load steps of the load-notch-strain curves, respectively. The FKM Guideline nonlinear specifies Newton’s methodfor this purpose. (Other solution methods would also be possible, but since this paperfocuses on the FKM Guideline nonlinear, only Newton’s method will be investigated, asspecified).

Newton’s method is a numerical solution method in which, starting from an initialvalue x0, the following iteration (iteration step n) is executed up to a termination criterion:

xn+1= xn −f(x n)

f′(x n) (11)

The objective is to find the root of the function f(x).In the FKM Guideline nonlinear [25], the start values of iterations ∆σ0 are defined to

be the corresponding elasticity-theoretical solutions ∆σel. The iteration is terminated aftera fixed number of ten steps.

To apply Newton’s method, Equation (11), the user must first determine the deriva-tives f’ of the function f from Equation (10).

With the value for the stresses ∆σel,pl determined in this way, the corresponding strainvalue can be calculated using Equation (5).

2.2. Notch Root Approximation according to Seeger and Beste

In addition to the extended version of Neuber’s rule, the notch root approximation ofSeeger and Beste [30–32] is utilized in [25]. The relationship applied in this case is shownin Equations (12) and (13).

εel,pl =σel,pl

( σelσel,pl

)2

· 2u2 · ln

(1

cos(u)

)− σelσel,pl

+1

·(ε∗el · E ·Kp

σel

)(12)

with

u =π

2·((σ el/σel,pl) − 1

Kp − 1

)(13)

The resulting root finding problem specified in the FKM Guideline nonlinear for thehysteresis branch is shown in Equations (14)–(16).

Appl. Sci. 2021, 11, 10339 7 of 23

f(∆σ el,pl) = 0 =

(∆σel,pl

E +2 ·(∆σel,pl

2K′

)1/n′)− ∆σel,pl

E ·[(

∆σel∆σel,pl

)2· 2

∆u2 · ln(

1cos(∆u)

)− ∆σel

∆σel,pl+1]·(

∆ε∗el·E·Kp∆σel

)(14)

with

∆u =π

2·((∆σ el/∆σel,pl)− 1

Kp − 1

)(15)

and

∆ε∗el =∆σel/ Kp

E+2(

∆σel/ Kp

2K′

)1/n′

(16)

for the application of Newton’s method, a starting value is required. However, whenconsidering the division term ∆u, it is noticeable that the starting value for the Newtoniteration cannot be the linearly elastically determined stress, as described in Section 2.1,since the quotient ∆σel/∆σel,pl = 1 and thus the division term ∆u = 0 would follow. As aresult, the undefined division by 0 would have to be carried out. Therefore, in [25], it issuggested to set the initial value ∆σ0 slightly lower depending on the limit load factor Kpaccording to Equation (17)

∆σ0= ∆σel ·(

1 −1− 1/Kp

1000

)(17)

analogous to the notch root approximation according to Neuber (Section 2.1), Newton’smethod and the termination criterion of 10 iteration steps are specified in [25] to solveEquations (14)–(16).

3. Issues concerning the Implementation of Notch Root Approximations

When implementing the two notch root approximations presented above in the fatiguestrength assessment in accordance with [25], the following questions arise for the user:

1. Is the formulation of the root finding problem a good option to ensure consistentaccuracy over the entire load range?

2. How can the derivative f’ of the functions for the notch root approximation used inNewton’s method be obtained? Is it better to perform the derivation analytically ornumerically?

3. Is the termination criterion specified in [25] for Newton’s method with a fixed numberof 10 iterations suitable for reliably performing the notch root approximations? Asalternative termination criteria, criteria based on an accuracy value could also beconsidered, which would lead to a variable number of iterations.

4. Are there other aspects to be considered with regard to a (numerically) stable andreliable implementation?

The following paragraphs first present proposed solutions for the mentioned aspects.These are later evaluated in Section 4 with regard to their effects on accuracy and efficiency.

3.1. Formulation of the Root Finding Problem for Use in Newton’s Method

In Section 2, the formulation of the root finding problem was described in the formdefined in [25]. The disadvantage of this formulation is that it optimizes the difference of thestrain (or strain range) εel,pl,RO calculated with the material behavior according to Rambergand Osgood (index RO) and εel,pl,NRA calculated with the notch root approximation (indexNRA) Equation (18).

f(σel,pl

)= 0 = εel,pl,RO(σ el,pl) − εel,pl,NRA(σ el,pl

)(18)

However, the underlying issue is that stresses and strains may be spread over severalorders of magnitude, e.g., 1 MPa to 1000 MPa. In such a case, optimization of a differenceleads to a situation where the relative error at the end of the iterations is dependent on

Appl. Sci. 2021, 11, 10339 8 of 23

the load level. It is thus considered reasonable to optimize on a factor so that a constantrelative accuracy is ensured over all orders of magnitude.

Therefore, as a possible alternative, a formulation will be suggested here in which thefactor between the resulting strain according to Equations (3)–(5), and the resulting strainfrom the notch root approximation is formed, Equation (19).

f(σel,pl

)= 0 =

εel,pl,RO

(σel,pl

)εel,pl,NRA

(σel,pl

)−1 (19)

For the case of the extended Neuber’s rule, Equation (20) is the result for the hysteresisbranch.

f(

∆σel,pl

)= 0 =

∆σel,plE +2 ·

(∆σel,pl

2·K′)1/n′

∆σel∆σel,pl

·Kp ·(

∆σel/KpE +2 ·

(∆σel/Kp

2·K′)1/n′

)−1 (20)

For the notch root approximation according to Seeger and Beste, the formulation viaquotient results in Equation (21) for the hysteresis branch.

f(

∆σel,pl

)= 0 =

∆σel,plE +2 ·

(∆σel,pl

2K′

)1/n′

∆σel,plE ·

[(∆σel

∆σel,pl

)2· 2

∆u2 · ln(

1cos(∆u)

)− ∆σel

∆σel,pl+1]· ∆ε∗el·E·Kp

∆σel

−1 (21)

using ∆u from Equation (15) and ∆ε∗el from Equation (16).

3.2. Derivatives of the Functions for Use in Newton’s Method

To apply Newton’s method, not only the functions for the root finding problem, i.e.,Equations (10), (14)–(16), (20) and (21), are required but also their derivatives need to beknown. In this section only the derivatives for the hysteresis branch formulations of theroot finding problems will be discussed, the full list of derivatives are given in Appendix B.

In the case of the extended notch root approximation according to Neuber, the deriva-tive of Equation (10) is relatively easy to determine analytically. It is given in Equation (22).

dfd∆σel,pl

= f′(

∆σel,pl

)=

1E+

(∆σel,pl

2·K′)1/n′−1

K′ · n′+

Kp · σel ·(

2 ·(

∆σel2·K′ ·Kp

)1/n′+ ∆σel

E·Kp

)∆σ2

el,pl(22)

The derivative for the alternative approach, Equation (20), described in the Section 3.1,is given in Equation (23).

dfd∆σel,pl

= f′(∆σ el,pl) =

∆σel,plE +2 ·

(∆σel,pl

2·K′)1/n′

Kp · ∆σel ·(

2 ·(

∆σel2·K′ ·Kp

)1/n′+ ∆σel

E·Kp

) +

∆σel,pl ·

1E +

(∆σel,pl

2·K′

)1/n′−1

K′ ·n′

Kp · ∆σel ·

(2 ·(

∆σel2·K′ ·Kp

)1/n′+ ∆σel

E·Kp

) (23)

The notch root approximation according to Seeger and Beste has clearly more extensiveexpressions in the root finding problem, namely, Equations (14)–(16) and (21). In principle,the derivatives can also be determined analytically. However, practically, the analyticalderivative can only be calculated using a computer algebra system (CAS). The derivativeof Equation (14) is shown in Equation (24).

Appl. Sci. 2021, 11, 10339 9 of 23

dfd∆σel,pl

= f′(∆σ el,pl) =1E +

(∆σel,pl

K′

)1/n′−1

K′ ·n′ − 1∆σel

Kp · ∆ε∗el ·

8∆σel2·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ2el,pl·π2·

(∆σel

∆σel,pl−1)2 − ∆σel

∆σel,pl+1

+Kp·∆ε∗el·∆σel,pl

∆σel

·

∆σel∆σ2

el,pl−

16∆σ2el·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ3el,pl·π2·

(∆σel

∆σel,pl−1)2 +

16∆σ3el·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ4el,pl·π2·

(∆σel

∆σel,pl−1)3 − 4∆σ3

el·sin(∆u)·(Kp−1)

∆σ4el,pl·π·cos(∆u)·

(∆σel

∆σel,pl−1)2

(24)

The derivative of Equation (21) is shown in Equation (25). It is noticeable that theseequations are practically difficult to handle without error.

dfd∆σel,pl

= f′(∆σ el,pl) =

∆σel·

1E+

(∆σel,pl

2K′

)1/n′−1

K′n′

Kp·∆ε∗ ·∆σel,pl·

8∆σ2el,pl ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 − ∆σel∆σel,pl

+1

−∆σel·

(∆σel,pl

E +2·(

∆σel,pl2K′

)1/n′)

Kp·∆ε∗ ·∆σ2el,pl·

8∆σ2el ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 −∆σel

∆σel,pl+1

− 1

Kp·∆ε∗ ·∆σel,pl·

8∆σ2el ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 −∆σel

∆σel,pl+1

2 · ∆σel ·

(∆σel,pl

E +2(∆σel,pl

2K′

)1/n′)

·

∆σel∆σel,pl

2−

16∆σ2el·ln

(1

cos(∆u)

)·(Kp−1)

2

π2·∆σ3el,pl·

(∆σel

∆σel,pl−1)2 +

16∆σ3el·ln

(1

cos(∆u)

)·(Kp−1)

2

π2·∆σ4el,pl·

(∆σel

∆σel,pl−1)3 − 4∆σ3

el·sin(∆u)·(Kp−1)

π·∆σ4el,pl·cos(∆u)·

(∆σel

∆σel,pl−1)2

(25)

Therefore, an alternative procedure will be discussed here. It is possible to approxi-mate the derivative numerically via the difference quotient, whereby the central differencequotient is used here, Equation (26). Here, h is the step size, which is set to 0.0001.

f′(x0) ≈f(x0+h)−f(x0−h)

2h(26)

3.3. Termination Criteria for Newton’s Method

Numerical methods generate an infinite sequence of values if no termination criterionis defined. If correctly chosen, the termination criterion guarantees that the procedureis terminated after a finite number of steps. In general, the procedure is terminated if adesired accuracy of the solution is reached or if no solution can be found with the selectedparameters. In the case of Newton’s method, the following variants are conceivable:

• Termination after a predefined number of iterations.• Termination when the value of the root finding problem falls below εV, i.e., when

Equation (27) is fulfilled. This is the case if the function value of the root findingproblem is sufficiently close to zero; compare Figure 4.

f(σn) < εV (27)

Appl. Sci. 2021, 11, 10339 10 of 23

Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 24

o εf = (20MPa/200,000MPa)+10-7(20MPa/200,000MPa)

= 1.001 or since in Equation (20) and Equation (21), 1 is subtracted. The result is ϵVq = 10-3.

Variants 2a and 2b are motivated by the assumption that Newton’s method with a quadratic order of convergence usually converges in 3 to 4 steps, meaning that it may not be necessary to use 10 steps in most cases, and therefore, on average, a performance gain might be achieved compared to variant 1, as proposed in [25].

Figure 4. Visualization of the Newton termination criterion “falling below a value”.

3.4. Consideration of the Numerical Stability In the case of the extended notch root approximation according to Neuber, no special

considerations have to be made with regard to stability. In all variants considered in this paper, the solution via Newton’s method reacts insensitively to small interferences of the input data.

The calculation of the notch root approximation according to Seeger and Beste, on the other hand, can result in complex numerical values, regardless of whether the analytical derivation or the numerical approximation is used. Complex values become possible in the context if, for example, the term ln(1/cos(u)), see Equation (12), becomes negative. This always seems to occur when a comparatively large number of Newton iterations are carried out, i.e., using the termination after 10 fixed steps. However, using termination criteria 2a and 2b, the condition for termination is always reached before complex values occur. A specific formula for determining when complex numbers are to be expected cannot be provided. The example in Table 1 demonstrates that a small rounding error can lead to unstable behavior. In this example, a small change in the 9th decimal place of the elastic stress range Δσ can lead to this unstable behavior and thus to complex values. This example is taken from the study described in the next section and thus represents a case that can also occur in this way in practical applications.

Table 1. Exemplary selection of parameters for which termination criteria 1 and 3 lead to complex values when applying the notch root approximation according to Seeger and Beste.

Parameter Value K’ in MPa 902.2915

n’ 0.187 Kp 1.7

Δσel leading to numerical stable behavior in MPa 110.022454508070

Figure 4. Visualization of the Newton termination criterion “falling below a value”.

In this paper, the following criteria are investigated in more detail:

1. Termination after 10 fixed steps, as suggested in [25].2. Termination after falling below a function value f(σn) < εV.

a. using f according to Equations (10) and (14) (i.e., finding the root of the differ-ence of εel,pl,RO−εel,pl,NRA) and εVd = 10−7.

b. using f according to Equations (20) and (21) (i.e., finding the root of the quotientof εel,pl,RO/εel,pl,NRA−1) and εVq= 10−3.

Variants 2a and 2b aim to reach a defined accuracy for the solution, while no predictioncan be made about the required number of iterations. On the other hand, the number ofiterations is fixed for variant 1, but no control of the calculation accuracy is provided.

The concrete threshold levels for termination criteria 2a and 2b were determined basedon the following considerations:

• A small remaining strain deviation in criterion 2a has a high influence, especially atlow stresses.

# For example, for steel (Young’s modulus E ≈ 200, 000 MPa), a value ofεVd= 10−7 leads to a deviation in the stress direction of 0.02 MPa at 1 MPa,which corresponds to a relative deviation of 2%.

# With increasing loads, the deviation of 0.02 MPa remains almost constant, sothe relative error decreases.

# The maximum error is ≈ 2%.

• If the absolute deviation of 0.02 MPa from the previous point is now applied to a loadheight of 20 MPa, then the following value results for εVq for termination criterion 2b:

# εf =(20 MPa/200,000 MPa)+10−7

(20 MPa/200,000 MPa) = 1.001 or since in Equations (20) and (21), 1 is

subtracted. The result is εVq= 10−3.

Variants 2a and 2b are motivated by the assumption that Newton’s method with aquadratic order of convergence usually converges in 3 to 4 steps, meaning that it may notbe necessary to use 10 steps in most cases, and therefore, on average, a performance gainmight be achieved compared to variant 1, as proposed in [25].

3.4. Consideration of the Numerical Stability

In the case of the extended notch root approximation according to Neuber, no specialconsiderations have to be made with regard to stability. In all variants considered in this

Appl. Sci. 2021, 11, 10339 11 of 23

paper, the solution via Newton’s method reacts insensitively to small interferences of theinput data.

The calculation of the notch root approximation according to Seeger and Beste, on theother hand, can result in complex numerical values, regardless of whether the analyticalderivation or the numerical approximation is used. Complex values become possible inthe context if, for example, the term ln(1/cos(u)), see Equation (12), becomes negative.This always seems to occur when a comparatively large number of Newton iterations arecarried out, i.e., using the termination after 10 fixed steps. However, using terminationcriteria 2a and 2b, the condition for termination is always reached before complex valuesoccur. A specific formula for determining when complex numbers are to be expectedcannot be provided. The example in Table 1 demonstrates that a small rounding error canlead to unstable behavior. In this example, a small change in the 9th decimal place of theelastic stress range ∆σel can lead to this unstable behavior and thus to complex values. Thisexample is taken from the study described in the next section and thus represents a casethat can also occur in this way in practical applications.

Table 1. Exemplary selection of parameters for which termination criteria 1 and 3 lead to complexvalues when applying the notch root approximation according to Seeger and Beste.

Parameter Value

K’ in MPa 902.2915n’ 0.187Kp 1.7

∆σel leading to numerical stable behavior in MPa 110.022454508070∆σel leading to numerical unstable behavior in

MPa 110.022454507071

4. Evaluation of Performance and Accuracy of Different Implementations

The described options in Section 3 regarding the implementation of the two notch rootapproximations are to be evaluated in terms of the following two aspects:

• The accuracy with which the notch root approximations are carried out affects theresults of the fatigue strength assessment.

• The calculation resources required for the application or the required computing timematter for an efficient implementation. When performing single calculations on asingle or a few individual locations of a component to be verified, this aspect is notimportant. However, in regard to performing many assessments, e.g., when appliedto a whole FE surface mesh of a component, the calculation time becomes a decisivefactor.

For this purpose, a large number of artificial examples are considered in the following,which contain all necessary input variables for a notch root approximation with the twonotch root approximations described above. For these examples, notch root approximationsare performed with the different implementation variants, and the calculation results ofthe different variants are compared with each other in terms of accuracy and efficiency.

4.1. Database

For the following evaluation of calculation quality and efficiency, a database with10 fictitious example geometries, each with 100 different configurations for the cyclicproperties, is used. These example geometries are represented by one dataset containingall required input values for the two notch root approximations specified in [25]. These areas follows:

• Maximum linear-elastic stress range relevant for the strength verification ∆σel,max.• Tensile strength Rm for estimation of Ramberg-Osgood parameters according to [25].• Limit load factor Kp.

These input values are defined within the value ranges given in Table 2.

Appl. Sci. 2021, 11, 10339 12 of 23

Table 2. Investigated parameter space for the applicability and performance of the notch approximation with differenttermination criteria when using Newton’s method.

Parameter Symbol Range Number ofEquidistant Steps in the Range

Limit load factor Kp 1–8 10Tensile strength Rm 300–1200 MPa 100

(resulting) Cyclic hardening coefficient K’ 640–3500 MPa (same as Rm)(resulting) Cyclic hardening exponent n’ 0.187 1

(resulting) Cyclic yield strength R′p0,2 200–1100 MPa (same as Rm)Maximum linear-elastic stress range ∆σel,max 0.1 · R′p0.2–0.8 · R′p0.2 ·Kp 100

The tensile strength Rm varies within a large range, representing the range of validitywithin the FKM Guideline nonlinear [25] for the steel material group and is used to estimatethe cyclic material properties by applying the method from [25]. The limit load factor isvaried within a range that is assumed to be relevant in practice. The maximum load isbased on the cyclic yield strength, which can be determined using Equation (4), and thelimit load factor, which, as already mentioned, is a measure of the maximum load that canbe applied.

The dataset is constructed as follows: Every possible permutation consisting of thefirst two rows of Table 2 is generated. Then, for each combination, the cyclic materialproperties shown in Rows 3 to 5 are estimated. Next, the maximum linear-elastic stressranges are determined based on the cyclic material properties in the range given in Table 2.Finally, for each of these combinations, notch root approximations on the hysteresis branch(since, as mentioned in Section 2, the initial load-notch-strain curve can be estimated fromthe load-notch-strain curve of the hysteresis branch) are performed according to [25] with200 discrete points. In summary, this leads to ~20,000,000 data points and notch rootapproximations.

4.2. Implementation into Software Code

To evaluate the abovementioned different variants of the notch root approximationimplementation, two different programming languages are utilized:

• The software environment MATLAB (version 2020b) was used to represent scriptlanguages and

• Fortran (Intel Ifort 2019 Update 5 using “-O3” optimization) was used to representcompiled languages in machine code.

Script languages and compiled programming languages can be optimized differentlyfor the respective calculation tasks. This in turn can lead to situations where a terminationcriterion for the Newton iteration, which results in high performance in one case, is notoptimal in the other case. For this reason, both of the abovementioned programminglanguages were used for this study.

All calculations were carried out separately for the variants of the termination criteria.For the notch root approximation of Seeger and Beste, in addition to the terminationcriterion, the type of derivation (analytical derivation according to Equations (24) and (25)or numerical approximation according to Equation (26)) is also varied here. Therefore, thefollowing nine variants are investigated:

1. Enhanced Neuber using the analytical derivative and termination after 10 fixed steps2. Enhanced Neuber using the analytical derivative, termination if Equation (10) falls

below εVd= 10−7

3. Enhanced Neuber using the analytical derivative, termination if Equation (20) fallsbelow εVq= 10−3

4. Seeger/Beste using the analytical derivative, termination after 10 fixed steps5. Seeger/Beste using the analytical derivative, termination if Equation (14) falls below

εVd= 10−7

Appl. Sci. 2021, 11, 10339 13 of 23

6. Seeger/Beste using the analytical derivative, termination if Equation (21) falls belowεVq= 10−3

7. Seeger/Beste using the numerical approximation of the derivative, termination after10 fixed steps

8. Seeger/Beste using the numerical approximation of the derivative, termination ifEquation (14) falls below εVd= 10−7

9. Seeger/Beste using the numerical approximation of the derivative, termination ifEquation (21) falls below εVq= 10−3

4.3. Results and Discussion

The database described in Section 4.1 is used to investigate the performance of theabovementioned variants. For each dataset, the notch root approximations of each of thenine variants 1 through 9 (Section 4.2) are executed using both MATLAB and Fortranimplementations.

For a qualitative comparison of the individual variants, two calculation runs areperformed. In the first calculation run, the number of Newton iterations until convergenceis recorded (which is independent from the programming language used). The distributionof the number of iterations is shown in Figure 5. The boxes graphically represent the 25%and 75% quantiles of the required number of Newton iteration steps to converge. In otherwords, 50% of all points are inside the box. The whiskers represent the minimum andmaximum values of iteration steps ignoring outliers.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 24

distribution of the number of iterations is shown in Figure 5. The boxes graphically represent the 25% and 75% quantiles of the required number of Newton iteration steps to converge. In other words, 50% of all points are inside the box. The whiskers represent the minimum and maximum values of iteration steps ignoring outliers.

The distribution of the number of iterations required until the termination criteria are met is independent of whether the analytical derivation or the numerical approximation is used. For this reason, Figure 5b shows the distribution of the required Newton steps independent of the method of derivation.

Figure 5. Number of Newton iterations required to reach the termination criteria when performing the extended Neuber notch root approximation (a) and the notch root approximation according to Seeger and Beste (b).

Figure 5a,b show that for termination criteria where an accuracy is specified, almost always less than 10 Newton steps are required. In some cases, i.e., when applying low loads, the termination criteria are fulfilled before the iteration starts. At the same time, it should be noted that in a few cases, when using a criterion quotient below ϵVq, more than 10 steps are required to achieve the desired accuracy.

In a second calculation run, only the calculation times are recorded. The resulting summed calculation time obtained by performing the extended version of Neuber’s rule on the ~20,000,000 data points, carried out on an Intel i5-8400H CPU, is on the order of 120–180 s when using MATLAB and 10–15 s when using Fortran. In the case of the MATLAB script, the overall fastest variant for the extended version of the Neuber rule is termination when Equation (20) (meaning the quotient of εel,pl,RO/εel,pl,NRA converges) reaches the desired accuracy. In case of the Fortran program the fastest variant is Equation (10) (difference εel,pl,RO-εel,pl,NRA) falls below ϵVd.

In the case of the notch root approximation of Seeger and Beste, the summed calculation time is on the order of 200–300 s when using MATLAB. In the Fortran program, the summed calculation time is on the order of 35–70 s. As in the case of the extended version of Neuber’s rule, Equation (21) (quotient εel,pl,RO/εel,pl,NRA) is the fastest approach when using MATLAB, and Equation (14) (difference εel,pl,RO-εel,pl,NRA) is the fastest approach when using Fortran.

The summed calculation times and the abovementioned findings are shown graphically in Figure 6.

0

5

10

15

10 S

teps

Diff

eren

cebe

low

V

d

Quo

tient

belo

w

Vq

0

5

10

15

Num

ber o

f ite

ratio

n st

eps

to c

onve

rgen

ce

Neuber(a)

Seeger & Beste(b)

Figure 5. Number of Newton iterations required to reach the termination criteria when performing the extended Neubernotch root approximation (a) and the notch root approximation according to Seeger and Beste (b).

The distribution of the number of iterations required until the termination criteria aremet is independent of whether the analytical derivation or the numerical approximationis used. For this reason, Figure 5b shows the distribution of the required Newton stepsindependent of the method of derivation.

Figure 5a,b show that for termination criteria where an accuracy is specified, almostalways less than 10 Newton steps are required. In some cases, i.e., when applying lowloads, the termination criteria are fulfilled before the iteration starts. At the same time, it

Appl. Sci. 2021, 11, 10339 14 of 23

should be noted that in a few cases, when using a criterion quotient below εVq, more than10 steps are required to achieve the desired accuracy.

In a second calculation run, only the calculation times are recorded. The result-ing summed calculation time obtained by performing the extended version of Neuber’srule on the ~20,000,000 data points, carried out on an Intel i5-8400H CPU, is on the or-der of 120–180 s when using MATLAB and 10–15 s when using Fortran. In the case ofthe MATLAB script, the overall fastest variant for the extended version of the Neuberrule is termination when Equation (20) (meaning the quotient of εel,pl,RO/εel,pl,NRA con-verges) reaches the desired accuracy. In case of the Fortran program the fastest variant isEquation (10) (difference εel,pl,RO−εel,pl,NRA) falls below εVd.

In the case of the notch root approximation of Seeger and Beste, the summed calcula-tion time is on the order of 200–300 s when using MATLAB. In the Fortran program, thesummed calculation time is on the order of 35–70 s. As in the case of the extended versionof Neuber’s rule, Equation (21) (quotient εel,pl,RO/εel,pl,NRA) is the fastest approach whenusing MATLAB, and Equation (14) (difference εel,pl,RO−εel,pl,NRA) is the fastest approachwhen using Fortran.

The summed calculation times and the abovementioned findings are shown graphi-cally in Figure 6.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 24

Figure 6. Summed calculation times for all 20,000,000 data points depending on the different formulations of the termination criteria for Newton’s method, the different formulations of the root finding problem and depending on the programming language (MATLAB and Fortran).

The distribution of the individual calculation times for each data point is shown in Figure 7. The calculation times for the extended version of Neuber’s rule are shown in Figure 7a. It can be seen that the termination after reaching a desired accuracy results in shorter calculation times on average compared to the 10-step approach.

The calculation times for the Seeger and Beste approach are shown in Figure 7b using the analytical derivation and in Figure 7c using the numerical approximation. As with Neuber’s rule, the approaches using a desired accuracy lead to faster calculations compared to the 10-step approach. This influence is greater in the implementation in Fortran by a factor of 2.5 than in MATLAB (Factor 2).

Using Figure 7b,c, the two approaches for the derivation can also be compared with regard to the computing time. Notably, in the case of the implementation in MATLAB, the approximate method for the derivative via the difference quotient leads to higher calculation times with the same termination criterion, while in Fortran, no significant influence is visible.

One attempt at an explanation is based on Fortran’s high degree of optimization. The recorded calculation times in Fortran are so short that other effects, such as background tasks in the operating system, have a greater influence than the actual calculations.

Considering that the numerical derivative from Equation (26) is significantly easier to handle than the analytical derivation from Equations (24) and (25), it can be concluded that for the notch root approximations according to Seeger and Beste, the numerical derivative in combination with the termination criterion 2a or 2b is a practical choice.

sum

med

com

puta

tion

time

fora

ll da

tapo

ints

[s]

10 S

teps

Diff

eren

cebe

low

ϵ VdQ

uotie

nt

belo

wϵ Vq

Diff

eren

cebe

low

ϵ Vd

Diff

eren

cebe

low

ϵ Vd

Quo

tient

be

low

ϵ Vq

Quo

tient

be

low

ϵ Vq

10 S

teps

10 S

teps

Analytical derivative

Numericalderivative

Analytical derivative

Neuber Seeger & Beste

102

101

103MatlabFortran

Figure 6. Summed calculation times for all 20,000,000 data points depending on the different formulations of the terminationcriteria for Newton’s method, the different formulations of the root finding problem and depending on the programminglanguage (MATLAB and Fortran).

The distribution of the individual calculation times for each data point is shown inFigure 7. The calculation times for the extended version of Neuber’s rule are shown in

Appl. Sci. 2021, 11, 10339 15 of 23

Figure 7a. It can be seen that the termination after reaching a desired accuracy results inshorter calculation times on average compared to the 10-step approach.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 24

The deviations of the estimated stresses of all variants in comparison to each other are on the order of 10 MPa. Deviations of this order of magnitude have a negligible influence on the calculated service life estimation.

It can therefore be concluded that for both notch root approximations in question, termination criteria using a specific accuracy significantly reduces the number of necessary Newton iterations and improves the computational performance compared to the 10-step termination criterion proposed in the FKM Guideline nonlinear [25]. At the same time, the resulting stresses and strains are not significantly influenced.

Figure 7. Calculation time of the various Newton termination criteria when using the extended notch root approximation according to Neuber (a), according to Seeger and Beste using the analytical derivative (b) and according to Seeger and Beste using the numerical approximation of the derivative (c).

Figure 7. Calculation time of the various Newton termination criteria when using the extended notch root approximationaccording to Neuber (a), according to Seeger and Beste using the analytical derivative (b) and according to Seeger and Besteusing the numerical approximation of the derivative (c).

The calculation times for the Seeger and Beste approach are shown in Figure 7b usingthe analytical derivation and in Figure 7c using the numerical approximation. As withNeuber’s rule, the approaches using a desired accuracy lead to faster calculations comparedto the 10-step approach. This influence is greater in the implementation in Fortran by afactor of 2.5 than in MATLAB (Factor 2).

Using Figure 7b,c, the two approaches for the derivation can also be compared withregard to the computing time. Notably, in the case of the implementation in MATLAB,

Appl. Sci. 2021, 11, 10339 16 of 23

the approximate method for the derivative via the difference quotient leads to highercalculation times with the same termination criterion, while in Fortran, no significantinfluence is visible.

One attempt at an explanation is based on Fortran’s high degree of optimization. Therecorded calculation times in Fortran are so short that other effects, such as backgroundtasks in the operating system, have a greater influence than the actual calculations.

Considering that the numerical derivative from Equation (26) is significantly easier tohandle than the analytical derivation from Equations (24) and (25), it can be concluded thatfor the notch root approximations according to Seeger and Beste, the numerical derivativein combination with the termination criterion 2a or 2b is a practical choice.

The deviations of the estimated stresses of all variants in comparison to each otherare on the order of 10−7 MPa. Deviations of this order of magnitude have a negligibleinfluence on the calculated service life estimation.

It can therefore be concluded that for both notch root approximations in question,termination criteria using a specific accuracy significantly reduces the number of necessaryNewton iterations and improves the computational performance compared to the 10-steptermination criterion proposed in the FKM Guideline nonlinear [25]. At the same time, theresulting stresses and strains are not significantly influenced.

5. Conclusions

In this paper, the theoretical background of notch root approximations is reviewedas far as it is necessary for practical implementation in the FKM Guideline nonlinear [25].The application of notch root approximations requires the solution of a nonlinear equation.This solution is usually performed using Newton’s method. The implementation of thismethod is the subject of this paper. In this context, the aspects of stability, performance,formulation of the root finding problem and accuracy are considered in more detail. This isinteresting, for example, if the notch root approximation is to be applied to each node inthe scope of variant calculations or in a postprocessing of a linear-elastic FE analysis.

The FKM Guideline nonlinear uses the extended Neuber’s rule and the notch rootapproximation according to Seeger and Beste. Both were examined more closely in thispaper with regard to the aspects mentioned above. The results obtained show that (notdepending on the notch root approximation method):

• 10 Newton steps, as suggested in [25], almost always lead to a desired accuracy butare not required in most cases.

• In the case of a script language, here represented by MATLAB, the formulation of theroot finding problem in the form of a quotient between the strain calculated via thematerial law and the strain calculated via the notch root approximation leads to thebest computational performance.

• In the case of a language compiled in machine code, here represented by Fortran,the formulation of the root finding problem in the form of a difference in the straincalculated via the material law and strain calculated via the notch root approximationleads to the best computational performance.

In particular, for the notch root approximation according to Seeger and Beste, thefollowing findings were also obtained:

• If Seeger and Beste’s notch root approximations are used and an incorrect terminationcriterion is selected, then there is a risk that the automated calculation will producecomplex numbers that cannot be interpreted meaningfully in terms of a service lifecalculation. Here, the use of the termination criteria with a desired accuracy leads tothe most stable solution, in the sense that no complex number arose in the examinedparameter field.

• The analytical derivative of the notch root approximation according to Seeger andBeste cannot be trivially used. The numerical approximation via the difference quotientis much easier to handle and leads to minor losses in performance when used inMATLAB and to comparable performance when used in Fortran.

Appl. Sci. 2021, 11, 10339 17 of 23

For implementation in engineering practice, the overall recommendation for bothnotch approximation methods considered here is therefore to use the termination criterionwhere the quotient of the strain εel,pl,RO/εel,pl,NRA converges, which leads to a constantrelative accuracy over all orders of magnitude, especially when using a script language.The source code implementing all considered variants in this study are provided as areference implementation in [43].

Author Contributions: Conceptualization, R.B., L.M., M.W. and A.E.; methodology, R.B. and M.W.;software, R.B.; validation, L.M., M.W. and A.E.; investigation, R.B.; resources, L.M.; writing—originaldraft preparation, R.B.; writing—review and editing, M.W., L.M. and A.E.; visualization, R.B. andM.W.; supervision, A.E.; project administration, M.W. and A.E. All authors have read and agreed tothe published version of the manuscript.

Funding: This research received no external funding. However, we acknowledge support from theOpen Access Publishing Fund of Clausthal University of Technology.

Data Availability Statement: The source code implementing all considered variants in this study isopenly available as MATLAB source in [43].

Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature

E Young’s modulusF function placeholder of the root finding problemf’ function placeholder of the derivative of the root finding problemFE finite elementFKM Abbreviation for the german research association “Forschungskuratorium Maschinenbau”H step sizek mean stress correction factorK′ cyclic hardening coefficientKp limit load factorLe reference load for local yieldingLp reference load for plastic collapsen′ cyclic hardening exponentNRA notch root approximationPRAJ load parameterPRAM load parameterR′p0.2 cyclic yield strengthRO Ramberg Osgood∆ range of stress/strain of a hysteresis∆εeff effective elastic-plastic strain range∆σeff effective elastic-plastic stress rangeεa strain amplitude of the detected stress-strain hysteresisεel linear-elastic local strainε∗el substitution strain for net section plasticityεel,pl elastic-plastic local strainεV placeholder for the convergence thresholdεVd convergence threshold when finding the root of the differenceεVq convergence threshold when finding the root of the quotientσ0 start value of elastic-plastic stress for the newton iterationσa stress amplitude of the detected stress-strain hysteresisσel linear-elastic local stressσel,pl elastic-plastic local stressσm mean stress of the detected stress-strain hysteresisσn value of elastic-plastic stress in the n-th newton iteration

Appl. Sci. 2021, 11, 10339 18 of 23

Appendix A. Derivation of the Root Finding Equations

In this section the derivation of the different root finding equations shall be describedin more detail. Here, the derivation in which the difference of εel,pl,RO according to Rambergand Osgood and εel,pl,NRA according to the notch root approximation is minimised takesplace first, followed by the derivation in which the quotient of both strains is minimised.The procedure is always demonstrated using the extended version of Neuber’s rule firstand then the results for the method according to Seeger and Beste are shown.

Appendix A.1. Root Finding Equations Formulated by Taking the Difference

The starting point is the extended version of the Neuber’s rule:

σel,pl · εel,pl = σel · ε∗el ·Kp (A1)

In a first step ε∗el, which considers the material behavior in case net section plasticitytakes place, needs to be replaced with the material behavior according to Ramberg andOsgood:

σel,pl · εel,pl= σel ·Kp ·(σel/Kp

E+

(σel/Kp

K′

)1/n′)

(A2)

Equation (A2) may be rearranged to:

εel,pl =σelσel,pl

·Kp ·(σel/Kp

E+

(σel/Kp

K′

)1/n′)

(A3)

In order to obtain a unique solution for the stress and strain, a second equation isrequired. As the FKM Guideline nonlinear uses Ramberg and Osgood material behaviorεel,pl from (A3) can be replaced with Equation (3):

σel,pl

E+

(σel,pl

K′

)1/n′

=σelσel,pl

·Kp ·(σel/Kp

E+

(σel/Kp

K′

)1/n′)

(A4)

The next step is to solve this equation for the desired elastic plastic stress. For thispurpose, the equation is converted into a root-finding problem by taking the difference ofthe strains.

0 =σel,pl

E+

(σel,pl

K′

)1/n′

− σelσel,pl

·Kp ·(σel/Kp

E+

(σel/Kp

K′

)1/n′)

(A5)

The steps described above can be applied for the hysteresis branch in the same way.In this case Equation (A6) results:

0 =∆σel,pl

E+2 ·

(∆σel,pl

2 ·K′)1/n′

− ∆σel∆σel,pl

·Kp ·(

∆σel/Kp

E+2 ·

(∆σel/Kp

2 ·K′)1/n′

)(A6)

Next, the identical procedure will be applied to the approach according to Seeger andBeste Equations (A7)–(A9).

εel,pl =σel,pl

( σelσel,pl

)2

· 2u2 · ln

(1

cos(u)

)− σelσel,pl

+1

·(ε∗el · E ·Kp

σel

)(A7)

with

u =π

2·((σ el/σel,pl)− 1

Kp − 1

)(A8)

Appl. Sci. 2021, 11, 10339 19 of 23

and

ε∗el =σel/ Kp

E+

(σel/ Kp

K′

)1/n′ (A9)

Equation (A7) (including (A8) and (A9)) is now equated with Equation (3).

σel,plE +

(σel,pl

K′

)1/n′=σel,pl

E ·[(

σelσel,pl

)2· 2

u2 · ln(

1cos(u)

)− σelσel,pl

+1]·(ε∗el·E·Kpσel

)(A10)

Rearranging the equation as already demonstrated for Neuber’s rule also leads hereto the difference formulation of the root finding problem.

0 =σel,pl

E+

(σel,pl

K′

)1/n′

−σel,pl

( σelσel,pl

)2

· 2u2 · ln

(1

cos(u)

)− σelσel,pl

+1

·(ε∗el · E ·Kp

σel

)(A11)

In the case of calculating a hysteresis branch Equation (A12) results:

0 =∆σel,pl

E+2 ·

(∆σel,pl

2K′

)1/n′

−∆σel,pl

( ∆σel∆σel,pl

)2

· 2∆u2 · ln

(1

cos(∆u)

)− ∆σel

∆σel,pl+1

·(∆ε∗el · E ·Kp

∆σel

)(A12)

with

∆u =π

2·((∆σ el/∆σel,pl) − 1

Kp′1

)(A13)

and

∆ε∗el =∆σel/ Kp

E+2(

∆σel/ Kp

2K′

)1/n′

(A14)

Appendix A.2. Root Finding Equations Formulated by Taking the Quotient

Instead of forming the difference between the strain from the material law and thestrain from the notch root approximation, the quotient can also be formed and convertedinto a zero-point formulation. In the case of the extended version of Neuber’s rule, startingfrom Equation (A4), the quotient can now be formed as follows.

1 =

σel,plE +

(σel,pl

K′

)1/n′

σelσel,pl

·Kp ·(σel/Kp

E +(σel/Kp

K′

)1/n′) (A15)

To obtain the root finding equation one has to subtract 1.

0 =

σel,plE +

(σel,pl

K′

)1/n′

σelσel,pl

·Kp ·(σel/Kp

E +(σel/Kp

K′

)1/n′) − 1 (A16)

The above steps can be applied in the same way for the hysteresis branch. In this caseEquation (A17) results:

0 =

∆σel,plE +2 ·

(∆σel,pl

2·K′)1/n′

∆σel∆σel,pl

·Kp ·(

∆σel/KpE +2 ·

(∆σel/Kp

2·K′)1/n′

) − 1 (A17)

Appl. Sci. 2021, 11, 10339 20 of 23

Transferred to the method according to Seeger and Beste, we obtain for the initial loadcurve

0 =

σel,plE +

(σel,pl

K′

)1/n′

σel,plE ·

[(σelσel,pl

)2· 2

u2 · ln(

1cos(u)

)− σelσel,pl

+1]· ε∗el·E·Kpσel

− 1 (A18)

and for the hysteresis branch Equation (A19)

0 =

∆σel,plE +2 ·

(∆σel,pl

2K′

)1/n′

∆σel,plE ·

[(∆σel

∆σel,pl

)2· 2

∆u2 · ln(

1cos(∆u)

)− ∆σel

∆σel,pl+1]· ∆ε∗el·E·Kp

∆σel

−1 (A19)

Appendix B. Derivatives of the Root Finding Equations

In this section the analytical derivatives of the different root finding equations, derivedin the Appendix A are listed.

In the case of the extended notch root approximation according to Neuber, the deriva-tives of Equation (A5) (initial load curve) and Equation (A6) (hysteresis branch) are givenin Equations (A20) and (A21).

f′(σel,pl

)=

1E+

(σel,pl

K′

)1/n′−1

K′ · n′+

Kp · σel ·((

σelK′ ·Kp

)1/n′+ σel

E·Kp

)σ2

el,pl(A20)

f′(

∆σel,pl

)=

1E+

(∆σel,pl

2·K′)1/n′−1

K′ · n′+

Kp · σel ·(

2 ·(

∆σel2·K′ ·Kp

)1/n′+ ∆σel

E·Kp

)∆σ2

el,pl(A21)

The derivatives for the alternative approach, i.e., Equation (A16) (initial load curve)and Equation (A17) are given in Equations (A22) and (A23).

f′(σel,pl

)=

σel,plE +

(σel,pl

K′

)1/n′

Kp · σel ·((

σelK′ ·Kp

)1/n′+ σel

E·Kp

) +

σel,pl ·

1E +

(σel,plK′

)1/n′−1

K′ ·n′

Kp · σel ·

((σel

K′ ·Kp

)1/n′+ σel

E·Kp

) (A22)

f′(∆σ el,pl) =

∆σel,plE +2·

(∆σel,pl

2·K′

)1/n′

Kp·∆σel·(

2·(

∆σel2·K′ ·Kp

)1/n′+

∆σelE·Kp

) +

∆σel,pl·

1E+

(∆σel,pl

2·K′

)1/n′−1

K′ ·n′

Kp·∆σel·

(2·(

∆σel2·K′ ·Kp

)1/n′+

∆σelE·Kp

) (A23)

In case of the notch root approximation according to Seeger and Beste the derivativeof the difference formulation of the root finding equation Equation (A11) (initial load curve)is shown in Equation (A24) and the derivative of Equation (A12) (hysteresis branch curve)in Equation (A25).

Appl. Sci. 2021, 11, 10339 21 of 23

dfdσel,pl

= f′(σel,pl

)= 1

E +

(σel,plK′

)1/n′−1

K′ ·n′ − 1σel·

Kp · ε∗el ·

8σ2el·ln

(1

cos(u)

)· (Kp−1)

2

σ2el,pl·π2·

(σelσel,pl

−1)2 − σel

σel,pl+1

−Kp·ε∗el·σel,plσel

·

σelσ2

el,pl−

16σ2el·ln

(1

cos(u)

)·(Kp−1)

2

σ3el,pl·π2·

(σelσel,pl

−1)2 +

16σ3el·ln

(1

cos(u)

)·(Kp−1)

2

σ4el,pl·π2·

(σelσel,pl

−1)3 − 4σ3

el·sin(u)·(Kp−1)

σ4el,pl·π·cos(u)·

(σelσel,pl

−1)2

(A24)

dfd∆σel,pl

= f′(∆σ el,pl) =1E +

(∆σel,pl

K′

)1/n′−1

K′ ·n′ − 1∆σel

Kp · ∆ε∗el ·

8∆σel2·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ2el,pl·π2·

(∆σel

∆σel,pl−1)2 − ∆σel

∆σel,pl+1

+Kp·∆ε∗el·∆σel,pl

∆σel

·

∆σel∆σ2

el,pl−

16∆σ2el·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ3el,pl·π2·

(∆σel

∆σel,pl−1)2 +

16∆σ3el·ln

(1

cos(∆u)

)·(Kp−1)

2

∆σ4el,pl·π2·

(∆σel

∆σel,pl−1)3 − 4∆σ3

el·sin(∆u)·(Kp−1)

∆σ4el,pl·π·cos(∆u)·

(∆σel

∆σel,pl−1)2

(A25)

In case of the notch root approximation according to Seeger and Beste the derivativeof the quotient formulation of the root finding problem Equation (A18) (initial load curve)is shown in Equation (A26) and of Equation (A19) (hysteresis branch curve and quotientformulation) in Equation (A27).

dfdσel,pl

= f′(σ el,pl) =

σel·

1E+

(σel,plK′

)1/n′−1

K′ ·n′

Kp·ε∗el·σel,pl·

8·σ2el ·ln

(1

cos(u)

)·(Kp−1)2

σ2 ·π2 ·(σelσel,pl

−1)2 − σel

σel,pl+1

−σel·

(σel,pl

E +(σel,pl

K′)1/n′

)

Kp·ε∗el·σ2el,pl·

8σ2el ·ln

(1

cos(u)

)·(Kp−1)2

σ2el,pl ·π

2 ·(σelσel,pl

−1)2 −

σelσel,pl

+1

− 1

Kp·ε∗el·σel,pl·

8σ2el ·ln

(1

cos(u)

)·(Kp−1)2

σ2el,pl ·π

2 ·(σelσel,pl

−1)2 −

σelσel,pl

+1

2 · σel ·

(σel,pl

E +(σel,pl

K′

)1/n′)

·

σelσ2

el,pl−

16σ2el·ln

(1

cos(u)

)·(Kp−1)

2

σ3el,pl·π2·

(σelσel,pl

−1)2 +

16σ3el·ln

(1

cos(u)

)·(Kp−1)

2

σ4el,pl·π2·

(σelσel,pl

−1)3 − 4σ3

el·sin(u)·(Kp−1)

σ4el,pl·π·cos(u)·

(σelσel,pl

−1)2

(A26)

Appl. Sci. 2021, 11, 10339 22 of 23

dfd∆σel,pl

= f′(∆σ el,pl) =

∆σel·

1E+

(∆σel,pl

2K′

)1/n′−1

K′n′

Kp·∆ε∗ ·∆σel,pl·

8∆σ2el,pl ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 − ∆σel∆σel,pl

+1

−∆σel·

(∆σel,pl

E +2·(

∆σel,pl2K′

)1/n′)

Kp·∆ε∗ ·∆σ2el,pl·

8∆σ2el ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 −∆σel

∆σel,pl+1

− 1

Kp·∆ε∗ ·∆σel,pl·

8∆σ2el ·ln

(1

cos(∆u)

)·(Kp−1)2

π2 ·∆σ2el,pl ·

(∆σel

∆σel,pl−1

)2 −∆σel

∆σel,pl+1

2 · ∆σel ·

(∆σel,pl

E +2(∆σel,pl

2K′

)1/n′)

·

∆σel∆σel,pl

2−

16∆σ2el·ln

(1

cos(∆u)

)·(Kp−1)

2

π2·∆σ3el,pl·

(∆σel

∆σel,pl−1)2 +

16∆σ3el·ln

(1

cos(∆u)

)·(Kp−1)

2

π2·∆σ4el,pl·

(∆σel

∆σel,pl−1)3 − 4∆σ3

el·sin(∆u)·(Kp−1)

π·∆σ4el,pl·cos(∆u)·

(∆σel

∆σel,pl−1)2

(A27)

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